| L(s) = 1 | + (0.800 + 0.599i)2-s + (2.81 − 1.05i)3-s + (0.281 + 0.959i)4-s + (−1.01 + 1.99i)5-s + (2.88 + 0.846i)6-s + (−0.890 − 0.193i)7-s + (−0.349 + 0.936i)8-s + (4.55 − 3.94i)9-s + (−2.00 + 0.989i)10-s + (−4.85 − 0.697i)11-s + (1.80 + 2.40i)12-s + (0.112 + 0.517i)13-s + (−0.597 − 0.689i)14-s + (−0.754 + 6.67i)15-s + (−0.841 + 0.540i)16-s + (−2.99 − 5.49i)17-s + ⋯ |
| L(s) = 1 | + (0.566 + 0.423i)2-s + (1.62 − 0.606i)3-s + (0.140 + 0.479i)4-s + (−0.452 + 0.891i)5-s + (1.17 + 0.345i)6-s + (−0.336 − 0.0732i)7-s + (−0.123 + 0.331i)8-s + (1.51 − 1.31i)9-s + (−0.634 + 0.313i)10-s + (−1.46 − 0.210i)11-s + (0.519 + 0.694i)12-s + (0.0312 + 0.143i)13-s + (−0.159 − 0.184i)14-s + (−0.194 + 1.72i)15-s + (−0.210 + 0.135i)16-s + (−0.727 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.21540 + 0.479729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.21540 + 0.479729i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.800 - 0.599i)T \) |
| 5 | \( 1 + (1.01 - 1.99i)T \) |
| 23 | \( 1 + (-2.84 + 3.86i)T \) |
| good | 3 | \( 1 + (-2.81 + 1.05i)T + (2.26 - 1.96i)T^{2} \) |
| 7 | \( 1 + (0.890 + 0.193i)T + (6.36 + 2.90i)T^{2} \) |
| 11 | \( 1 + (4.85 + 0.697i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.112 - 0.517i)T + (-11.8 + 5.40i)T^{2} \) |
| 17 | \( 1 + (2.99 + 5.49i)T + (-9.19 + 14.3i)T^{2} \) |
| 19 | \( 1 + (-3.07 + 0.901i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (0.421 - 1.43i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.23 - 7.07i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.415 - 5.80i)T + (-36.6 - 5.26i)T^{2} \) |
| 41 | \( 1 + (-0.940 + 1.08i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.16 - 11.1i)T + (-32.4 + 28.1i)T^{2} \) |
| 47 | \( 1 + (0.416 - 0.416i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.324 + 1.49i)T + (-48.2 - 22.0i)T^{2} \) |
| 59 | \( 1 + (7.41 - 11.5i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.26 + 2.40i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-2.31 + 3.09i)T + (-18.8 - 64.2i)T^{2} \) |
| 71 | \( 1 + (-0.108 - 0.753i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.0520 - 0.0284i)T + (39.4 + 61.4i)T^{2} \) |
| 79 | \( 1 + (13.7 + 8.80i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.83 - 0.417i)T + (82.1 + 11.8i)T^{2} \) |
| 89 | \( 1 + (-2.73 + 5.99i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-6.39 + 0.457i)T + (96.0 - 13.8i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.71598251177029289032742494241, −11.48782596789695233206615896221, −10.26300896346339750840494443166, −9.039886844741688768244125685413, −8.039936043358914893334778042347, −7.32293553301139804599388029481, −6.59202018094411160036381359557, −4.72991559468434336405173808152, −3.08936713396127114798468041350, −2.74074648020749611654462935948,
2.19513657742356317100546671656, 3.45267422350098613744982258033, 4.35986497152675152618960852645, 5.51213404072315407360297377893, 7.54683973844211234981196291390, 8.294184861556344121680787329792, 9.268897416833266252110275404941, 10.05799353114214232528196919630, 11.09131862130627867388626769145, 12.57678609851820231384190106765
